Trans-heteroclinic bifurcation : a novel type of catastrophic shift

Global and local bifurcations are extremely important since they govern the transitions between different qualitative regimes in dynamical systems. These transitions or tipping points, which are ubiquitous in nature, can be smooth or catastrophic. Smooth transitions involve a continuous change in the steady state of the system until the bifurcation value is crossed, giving place to a second-order phase transition. Catastrophic transitions involve a discontinuity of the steady state at the bifurcation value, giving place to first-order phase transitions. Examples of catastrophic shifts can be found in ecosystems, climate, economic or social systems. Here we report a new type of global bifurcation responsible for a catastrophic shift. This bifurcation, identified in a family of quasi-species equations and named as trans-heteroclinic bifurcation, involves an exchange of stability between two distant and heteroclinically connected fixed points. Since the two fixed points interchange the stability without colliding, a catastrophic shift takes place. We provide an exhaustive description of this new bifurcation, also detailing the structure of the replication-mutation matrix of the quasi-species equation giving place to this bifurcation. A perturbation analysis is provided around the bifurcation value. At this value the heteroclinic connection is replaced by a line of fixed points in the quasi-species model. But it is shown that, if the replication-mutation matrix satisfies suitable conditions, then, under a small perturbation, the exchange of heteroclinic connections is preserved, except on a tiny range around the bifurcation value whose size is of the order of magnitude of the perturbation. The results presented here can help to understand better novel mechanisms behind catastrophic shifts and contribute to a finer identification of such transitions in theoretical models in evolutionary biology and other dynamical systems

Activation of effector immune cells promotes tumor stochastic extinction: A homotopy analysis approach

In this article we provide homotopy solutions of a cancer nonlinear model describing the dynamics of tumor cells in interaction with healthy and effector immune cells. We apply a semi-analytic technique for solving strongly nonlinear systems - the Step Homotopy Analysis Method (SHAM). This algorithm, based on a modification of the standard homotopy analysis method (HAM), allows to obtain a one-parameter family of explicit series solutions. By using the homotopy solutions, we first investigate the dynamical effect of the activation of the effector immune cells in the deterministic dynamics, showing that an increased activation makes the system to enter into chaotic dynamics via a period-doubling bifurcation scenario. Then, by adding demographic stochasticity into the homotopy solutions, we show, as a difference from the deterministic dynamics, that an increased activation of the immune cells facilitates cancer clearance involving tumor cells extinction and healthy cells persistence. Our results highlight the importance of therapies activating the effector immune cells at early stages of cancer progression

On chaos, transient chaos and ghosts in single populations models with allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)(infinity) occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects. (C) 2011 Elsevier Ltd. All rights reserved

Dynamical effects of loss of cooperation in discrete-time hypercycles

Hypercycles' dynamics have been widely investigated in the context of origins of life, especially using time-continuous dynamical models. Different hypercycle architectures jeopardising their stability and persistence have been discussed and investigated, namely the catalytic parasites and the short-circuits. Here we address a different scenario considering RNA-based hypercycles in which cooperation is lost and catalysis shifts to density-dependent degradation processes due to the acquisition of cleaving activity by one hypercycle species. That is, we study the dynamical changes introduced by a functional shift. To do so we use a discrete-time model that can be approached to the time continuous limit by means of a temporal discretisation parameter, labelled $C$. We explore dynamical changes tied to the loss of cooperation in two-, three-, and fourmember hypercycles in this discrete-time setting. With cooperation, the all-species coexistence in two- and three-member hypercycles is governed by an internal stable fixed point. When one species shifts to directed degradation, a transcritical bifurcation takes place and the other hypercycle members go to extinction. The asymptotic dynamics of the four-member system is governed by an invariant curve in its cooperative regime. For this system, we have identified a simultaneous degenerate transcritical-NeimarkSacker bifurcation as cooperation switches to directed degradation. After these bifurcations, as we found for the other systems, all the cooperative species except the one performing degradation become extinct. Finally, we also found that the observed bifurcations and asymptotic dynamical behaviours are independent of $C$. Our results can help in understanding the impact of changes in ecological interactions (i.e., functional shifts) in multi-species systems and to determine the nature of the transitions tied to coextinctions and out-competition processes in both ecosystems and RNA-based systems

Quasispecies Spatial Models for RNA Viruses with Different Replication Modes and Infection Strategies

Empirical observations and theoretical studies suggest that viruses may use different replication strategies to amplify their genomes, which impact the dynamics of mutation accumulation in viral populations and therefore, their fitness and virulence. Similarly, during natural infections, viruses replicate and infect cells that are rarely in suspension but spatially organized. Surprisingly, most quasispecies models of virus replication have ignored these two phenomena. In order to study these two viral characteristics, we have developed stochastic cellular automata models that simulate two different modes of replication (geometric vs stamping machine) for quasispecies replicating and spreading on a two-dimensional space. Furthermore, we explored these two replication models considering epistatic fitness landscapes (antagonistic vs synergistic) and different scenarios for cell-to-cell spread, one with free superinfection and another with superinfection inhibition. We found that the master sequences for populations replicating geometrically and with antagonistic fitness effects vanished at low critical mutation rates. By contrast, the highest critical mutation rate was observed for populations replicating geometrically but with a synergistic fitness landscape. Our simulations also showed that for stamping machine replication and antagonistic epistasis, a combination that appears to be common among plant viruses, populations further increased their robustness by inhibiting superinfection. We have also shown that the mode of replication strongly influenced the linkage between viral loci, which rapidly reached linkage equilibrium at increasing mutations for geometric replication. We also found that the strategy that minimized the time required to spread over the whole space was the stamping machine with antagonistic epistasis among mutations. Finally, our simulations revealed that the multiplicity of infection fluctuated but generically increased along time.This work has been funded by the Human Frontier Science Program Organization Grant RGP12/2008 and the Spanish Ministerio de Ciencia e Innovacio´n Grant BFU2009-06993. The authors also acknowledge support from the Santa Fe Institute. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Peer reviewe

On Dynamics and Invariant Sets in Predator-Prey Maps

A multitude of physical, chemical, or biological systems evolving in discrete time can be modelled and studied using difference equations (or iterative maps). Here we discuss local and global dynamics for a predator-prey two-dimensional map. The system displays an enormous richness of dynamics including extinctions, co-extinctions, and both ordered and chaotic coexistence. Interestingly, for some regions we have found the so-called hyperchaos, here given by two positive Lyapunov exponents. An important feature of biological dynamical systems, especially in discrete time, is to know where the dynamics lives and asymptotically remains within the phase space, that is, which is the invariant set and how it evolves under parameter changes. We found that the invariant set for the predator-prey map is very sensitive to parameters, involving the presence of escaping regions for which the orbits go out of the domain of the system (the species overcome the carrying capacity) and then go to extinction in a very fast manner. This theoretical finding suggests a potential dynamical fragility by which unexpected and sharp extinctions may take place

Dynamics of alternative modes of RNA replication for positive-sense RNA viruses

We propose and study nonlinear mathematical models describing the intracellular time dynamics of viral RNA accumulation for positive sense single-stranded RNA viruses. Our models consider different replication modes ranging between two extremes represented by the geometric replication (GR) and the linear stamping machine replication (SMR). We first analyze a model that quantitatively reproduced experimental data for the accumulation dynamics of both polarities of Turnip mosaic potyvirus RNAs. We identify a non-degenerate transcritical bifurcation governing the extinction of both strands depending on three key parameters: the mode of replication (®), the replication rate (r) and the degradation rate (±) of viral strands. Our results indicate that the bifurcation associated with ® generically takes place when the replication mode is closer to the SMR, thus suggesting that GR may provide viral strands with an increased robustness against degradation. This transcritical bifurcation, which is responsible for the switching from an active to an absorbing regime, suggests a smooth (i.e., second-order), absorbing-state phase transition. Finally, we also analyze a simplified model that only incorporates asymmetry in replication tied to differential replication modes.This work has been funded by the Human Frontier Science Program Organization grant RGP12/2008, by the Spanish Ministerio de Ciencia e Innovaci´on grants BIO2008-01986 (JAD) and BFU2009-06993 (SFE), and by the Santa Fe Institute. FM is the recipient of a predoctoral fellowship from Universitat Polit`ecnica de Val`encia. We also want to thank the hospitality and support of the Kavli Institute for Theoretical Physics (University of California at Santa Barbara), where part of this work was developed (grant NSF PHY05-51164).Peer reviewe

Noise-induced bistability in the quasi-neutral coexistence of viral RNAs under different replication modes

[EN] Evolutionary and dynamical investigations into real viral populations indicate that RNA replication can range between the two extremes represented by so-called 'stamping machine replication' (SMR) and 'geometric replication' (GR). The impact of asymmetries in replication for single-stranded (+) sense RNA viruses has been mainly studied with deterministic models. However, viral replication should be better described by including stochasticity, as the cell infection process is typically initiated with a very small number of RNA macromolecules, and thus largely influenced by intrinsic noise. Under appropriate conditions, deterministic theoretical descriptions of viral RNA replication predict a quasi-neutral coexistence scenario, with a line of fixed points involving different strands' equilibrium ratios depending on the initial conditions. Recent research into the quasi-neutral coexistence in two competing populations reveals that stochastic fluctuations fundamentally alter the mean-field scenario, and one of the two species outcompetes the other. In this article, we study this phenomenon for viral RNA replication modes by means of stochastic simulations and a diffusion approximation. Our results reveal that noise has a strong impact on the amplification of viral RNAs, also causing the emergence of noise-induced bistability. We provide analytical criteria for the dominance of (+) sense strands depending on the initial populations on the line of equilibria, which are in agreement with direct stochastic simulation results. The biological implications of this noise-driven mechanism are discussed within the framework of the evolutionary dynamics of RNA viruses with different modes of replication.The research leading to these results has received funding from 'la Caixa' Foundation. 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